An apparent paradox in the mechanics of strings

Capriz, Gianfranco; Mazzini, Giovanni

doi:10.1007/bf01020319

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2011

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“…In recent years the problem has been debated in the physics literature, in particular the issue of whether energy is conserved and whether the tip of the chain falls at an acceleration equal to gravity or faster [7,6,8,10,16,17,27,35,39,40,37]. See Wong and Yasui [42] or McMillen [23] for a good survey of the literature.…”

Section: Introductionmentioning

confidence: 99%

The motion of whips and chains

Preston

2011

Journal of Differential Equations

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We study the motion of an inextensible string (a whip) fixed at one point in the absence of gravity, satisfying the equations $$ \eta_{tt} = \partial_s(\sigma \eta_s), \qquad \sigma_{ss}-\lvert \eta_{ss}\rvert^2 = -\lvert \eta_{st}\rvert^2, \qquad \lvert \eta_s\rvert^2 \equiv 1 $$ with boundary conditions $\eta(t,1)=0$ and $\sigma(t,0)=0$. We prove local existence and uniqueness in the space defined by the weighted Sobolev energy $$ \sum_{\ell=0}^m \int_0^1 s^{\ell} \lvert \partial_s^{\ell}\eta_t\rvert^2 \, ds + \int_0^1 s^{\ell+1} \lvert \partial_s^{\ell+1}\eta\rvert^2 \, ds, $$ when $m\ge 3$. In addition we show persistence of smooth solutions as long as the energy for $m=3$ remains bounded. We do this via the method of lines, approximating with a discrete system of coupled pendula (a chain) for which the same estimates hold.Comment: 47 pages, 8 figure

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Section: Introductionmentioning

confidence: 99%

The motion of whips and chains

Preston

2011

Journal of Differential Equations

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An apparent paradox in the mechanics of strings

Cited by 1 publication

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The motion of whips and chains

The motion of whips and chains

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